Products of Coplanar Vectors

Excerpts

• Products of Coplanar Vectors

  Its geometrical meaning is the product of $A$ and the orthogonal projection of $B$ upon $A$.

• Products of Coplanar Vectors

  The geometrical meaning of $\mathrm{S}AB$ is the product of $A$ and the orthogonal projection of $B$ upon $A$.

• Products of Coplanar Vectors

  By the reciprocal of a vector is meant the vector which combined with the original vector produces the product $+1$.

• Products of Coplanar Vectors

  A common explanation which is given of $ij = k$ is that $i$ is an operator, $j$ an operand, and $k$ the result. The kind of operator which $i$ is supposed to denote is a quadrant of turning round the axis $i$; it is supposed not to be an axis, but a quadrant of rotation round an axis. This explains the result $ij = k$, but unfortunately it does not explain $ii = +$; for it would give $ii = i$.

• Products of Coplanar Vectors

  the whole kinetic energy is obtained, not by vector, but by simple addition, when the components are rectangular.

• Products of Coplanar Vectors

  It follows that $\mathrm{V}BA = -\mathrm{V}AB$. It is to be observed that the coordinates of $A$ and $B$ are mere component vectors, whereas $A$ and $B$ themselves are taken in a real order.

• Products of Coplanar Vectors

  We assume that their product is obtained by applying the distributive law, but we do not assume that the order of the factors is indifferent.

Equations

• Products of Coplanar Vectors

  A = a_1i + a_2j

  Any vector confined to the plane is the sum of two rectangular components, taken along the directions i and j.

• Products of Coplanar Vectors

  AB = (a_1i + a_2j)(b_1i+b_2j) = a_1b_1ii + a_2b_2jj + a_1b_2ij + a_2b_2ji

  The product of two coplanar vectors, expanded by the distributive law without assuming the factors commute. The book's last term reads a_2b_2ji, which appears to be a typo for a_2b_1ji, since the following reduction requires b_1.

• Products of Coplanar Vectors

  ij = k

  The product of two directions at right angles is the direction normal to both, an assumption the book adopts as suggested by ordinary algebra.

• Products of Coplanar Vectors

  AB = a_1b_1 + a_2b_2 + (a_1b_2 - a_2b_1)k

  The complete product of two coplanar vectors splits into a scalar partial product, independent of direction, and a vector partial product along the normal k.

• Products of Coplanar Vectors

  OPQ = a_1 b_2 - \frac{1}{2} a_2 a_2 - \frac{1}{2} b_1 b_2 - \frac{1}{2} (a_1 - b_1)(b_2 - a_2) = \frac{1}{2}(a_1 b_2 - a_2 b_1)

  The area of triangle OPQ, formed by the vectors A and B, equals half the magnitude of the vector product. Flag: the middle term of the book's decomposition reads a_2 a_2, which looks like a typo (likely a_2 b_2 or similar); the final expression is what the book uses and is stated as the result.

• Products of Coplanar Vectors

  \mathrm{V}BA = -\mathrm{V}AB

  The vector product changes sign when the order of the two factors is reversed.

• Products of Coplanar Vectors

  \mathrm{S}BA = \mathrm{S}AB

  The scalar product is symmetric: interchanging the two vectors leaves it unchanged, since it is the product of one vector with the projection of the other.

• Products of Coplanar Vectors

  A^2 = a_1^2 + a_2^2 = a^2

  The square of a vector is its squared magnitude, independent of direction, and so is positive.

• Products of Coplanar Vectors

  \mathrm{S}AB = ab \cos \alpha\beta

  The scalar product equals the product of the magnitudes a and b with the cosine of the angle between the directions alpha and beta.

• Products of Coplanar Vectors

  \mathrm{V}AB = ab \sin \alpha\beta \cdot \overline{\alpha\beta}

  The vector product has magnitude ab times the sine of the angle between alpha and beta, and points along the direction normal to both, in the sense of the right-handed screw.

• Products of Coplanar Vectors

  A^{-1} = \frac{1}{a}\alpha = \frac{a\alpha}{a^2} = \frac{a_1i + a_2j}{a_1^2 + a_2^2}

  The reciprocal of a vector has the same direction and the reciprocal of its magnitude, and its components are the original components divided by the sum of their squares.

• Products of Coplanar Vectors

  A^{-1}B = \frac{1}{a^2}AB

  The product of the reciprocal of A with B is the complete product of A and B divided by the square of A's magnitude.

• Products of Coplanar Vectors

  \mathrm{S}A^{-1}B = \dfrac{b}{a}\cos \alpha\beta

  The scalar product of the reciprocal of A with B is b/a times the cosine of the angle between A and B.

• Products of Coplanar Vectors

  \mathrm{V}A^{-1}B = \dfrac{b}{a} \sin \alpha\beta \cdot \overline{\alpha\beta}

  The vector product of the reciprocal of A with B is b/a times the sine of the angle between A and B, directed along their pole.

• Products of Coplanar Vectors

  \mathrm{V}(A + B)C = \mathrm{V}AC + \mathrm{V}BC

  The vector product distributes over a sum: the product of a sum of vectors with C equals the sum of the separate products.

• Products of Coplanar Vectors

  (a_1b_1 + a_2b_2)(c_1i + c_2j) + (a_1b_2 - a_2b_1)(-c_2i + c_1j)

  The product of three coplanar vectors (AB)C expands into a scalar-times-C term plus a vector-product term times the complementary vector of C. The book's relation is written over two lines and this is its second line, without the leading equals sign.

• Products of Coplanar Vectors

  \mathrm{S}BC \cdot A + \mathrm{V}A(\mathrm{V}BC)

  The product A(BC) of three coplanar vectors equals the scalar product of B and C times A, plus the vector product of A with the vector product of B and C; the latter depends on the mode of association.

• Products of Coplanar Vectors

  a^2 + b^2 + 2ab\cos\alpha\beta

  The square of the sum of two non-successive vectors, in magnitudes, is the sum of their squares plus twice the product of magnitudes and the cosine of the angle between them.

• Products of Coplanar Vectors

  A^2+B^2 + 2\mathrm{S}AB + 2\mathrm{V}AB

  The square of the sum of successive vectors has a scalar part giving the squared third side and a vector part giving twice the vector product.

• Products of Coplanar Vectors

  A^2 + B^2 + C^2 + 2AB + 2AC + 2BC

  The square of the sum of three successive vectors, with product terms formed in the order of the vectors in the trinomial.

• Products of Coplanar Vectors

  a^2 + b^2 + c^2 + 2ab\cos \alpha\beta + 2ac\cos \alpha\gamma + 2bc\cos\beta\gamma

  The scalar part of the square of a trinomial of coplanar successive vectors: the squared magnitudes plus twice each pairwise product of magnitudes times the cosine of the angle between them.

• Products of Coplanar Vectors

  \{ 2ab\sin\alpha\beta + 2ac\sin\alpha\gamma + 2bc\sin\beta\gamma\} \cdot \overline{\alpha\beta}

  The vector part of the square of a trinomial of coplanar successive vectors: twice the sum of the pairwise vector products, along the normal to the plane.

Problems